Research Article | Open Access

# On Fixed Point Theory of Monotone Mappings with Respect to a Partial Order Introduced by a Vector Functional in Cone Metric Spaces

**Academic Editor:**Micah Osilike

#### Abstract

We presented some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to a partial order introduced by a vector functional in cone metric spaces. In addition, we proved not only the existence of maximal and minimal fixed points but also the existence of the largest and the least fixed points of single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the recent results.

#### 1. Introductions

Throughout this paper, let be a complete cone metric space over a total minihedral and continuous cone of a normed vector space . A vector functional introduces a partial order on as follows: for all , where is the partial order on determined by the cone . Using the partial order introduced by the vector functional , Agarwal and Khamsi [1] extended Caristi's fixed point theorem [2] to the case of cone metric space and proved that all mapping (resp., ) such that has a fixed point provided that is lower semicontinuous and bounded below on . In [1, 3], the authors studied Kirk's problem [4, 5] in the case of cone metric spaces and obtained some generalized Caristi's fixed point theorems in cone metric spaces. For the researches on the generalization of primitive Caristi's result in the case of metric spaces, we refer the readers to [6–12]. For other references concerned with various fixed point results for one, two, three, or four self-mappings in the setting of metric, ordered metric, partial metric, Prešić-type mappings, cone metric, G-metric spaces, and so forth, we refer the readers to [13–24].

In particular, when , the partial order defined by (1) is reduced to the one defined by Caristi [2] who denote it by . Zhang [25, 26] and Li [27] considered the existence of fixed points of a mapping (resp., ) such that for some , and proved some maximal and minimal fixed point theorems at the expense that is monotone with respect to the partial order .

In this paper, we shall extend the results of Zhang [25, 26] and Li [27] to the case of cone metric spaces. Some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to the partial order are established in cone metric spaces. In addition, not only the existence of maximal and minimal fixed points but also the existence of largest and least fixed points is proved for single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the results of Zhang [25] and Li [27].

#### 2. Preliminaries

First, we recall some definitions and properties of cones and cone metric spaces; these can be found in [1, 3, 17–24, 28–30].

Let be a topological vector space. A cone of is a nonempty closed subset of such that for all and all , and , where is the zero element of . A cone of determines a partial order on by for all . For all with , we write , where is the interior of .

Let be a cone of a topological vector space. is total order minihedral [29] if, for all upper bounded nonempty total ordered subset of , exists in . Equivalently, is total order minihedral if, for all lower bounded nonempty total ordered subset of , exists in .

Let be a normed vector space. A cone of is continuous [1, 3] if, for all subset of , exists implies , and exists implies . A cone of is normal [30] if there exists such that for all , implies , and the minimal is called a normal constant of . Equivalently, A cone of is normal provided that for all with for all , and imply for some .

*Remark 1. * A total order minihedral cone of a normed space is certainly normal see [29].

Let be a nonempty set and a cone of a topological vector space . A cone metric [28] is a mapping such that for all , if and only if , , .

A pair is called a cone metric space over if is a cone metric. Let be a cone metric space over a cone of a topological vector space . A sequence in converges [28] to (denote ) if, for all with , there exists a positive integer such that for all . A sequence in is Cauchy [28] if, for all with , there exists a positive integer such that for all . A cone metric space is complete [28] if all Cauchy sequence in converges to a point . A vector functional is sequentially continuous at some if for all such that . If, for all , is sequentially continuous at , then is sequentially continuous.

*Remark 2. *Let be a cone metric space over a normal cone of a normed vector space and a sequence in . Then if and only if , and is Cauchy if and only if see [28].

Let be a nonempty set and a partial order on . For all with , set , , and . Let be a nonempty subset of . A set-valued mapping is increasing on if, for all with and all , there exists such that . A set-valued mapping is quasi-increasing if, for all with and all , there exists such that . In particular, a single-valued mapping is increasing on if, for all with , .

A point is called a fixed point of a set-valued (resp., single-valued) mapping if (resp. ). Let be a nonempty subset of and let be a fixed point of a mapping . is called a maximal (resp. minimal) fixed point of in if for all fixed point of , (resp., ) implies . is called a largest (resp., least) fixed point of in if, for all fixed point of , (resp., ). A largest (resp., least) fixed point of in is naturally a maximal (resp., minimal) fixed point in , but the converse may not be true.

#### 3. Fixed Point Theorems

In this section, we always assume that the partial order is defined by (1).

Theorem 3. * Let be a complete partially ordered cone metric space over a total order minihedral and continuous cone of a normed vector space . Let be a sequentially continuous vector functional and let be a set-valued mapping such that is compact for all . Assume that there exists such that is bounded below on , is increasing on , and . Then has a maximal fixed point .*

*Proof. *Since is a total order minihedral cone and is a normed space, then is a normal cone by Remark 1. Set
Clearly, is nonempty since . Let be an increasing chain, where is a directed set. Then by (1) we have
for all with . This implies that is a decreasing chain in . Since is total order minihedral and is bounded below on , then exists in . Moreover, since is continuous. Therefore there exists an increasing sequence such that , that is,
By (1) we have for all such that ,
Let , by (6) we have and hence by the normality of . Moreover by Remark 2, is a Cauchy sequence in . Therefore by the completeness of , there exists some such that
Note that is an increasing sequence of , then by (1), we have for all ,
And, for all ,
where is an arbitrary integer. Let , then by (8) and the continuity of we have and , that is, and . Moreover the arbitrary property of forces that
for all . By , there exists such that
for all . Since is increasing on , then by (11) and , there exists such that
for all . This together with (12) implies that
for all . Note that is compact, and there exists a subsequence and such that
From (14) we have for all and hence by (1),
for all . Let , then by (8), (15), and the continuity of we have , that is, . This implies that and hence by .

For all , if there exists some such that , then is an upper bound of by (11). Otherwise, there exists some such that for all . Thus by (1) we have for all . Let , by (6) we have ; that is, . So we have and hence for all . Note that is a decreasing chain, then for all . Moreover for all since is an increasing chain. Hence has an upper bound in . By Zorn's lemma, has a maximal element ; that is, for all , implies . By , there exists such that . Moreover by the increasing property of on , there exists such that . Thus we have by . This indicates and hence . Finally the maximality of in forces that ; that is, is a maximal fixed point of in . The proof is complete.

Theorem 4. * Let be a complete partially ordered cone metric space over a total order minihedral and continuous cone of a normed vector space . Let be a sequentially continuous vector functional and be a set-valued mapping such that is compact for all . Assume that there exists such that is bounded above on , is quasi-increasing on , and . Then has a minimal fixed point .*

*Proof. *Set
Clearly, . By the same method used in the proof of Theorem 3, we can prove that has a minimal element which is also a minimal fixed point of in . The proof is complete.

*Remark 5. * If is a single-valued mapping, then is naturally compact for all . Hence both of Theorems 3 and 4 are still valid for a single-valued mapping.

In particular when is a single-valued mapping, we have the following further results.

Theorem 6. *Let be a complete partiallly ordered cone metric space over a total order minihedral and continuous cone of a normed vector space . Let be a sequentially continuous vector functional and let be a single-valued mapping. Assume that there exists such that is bounded below on , is increasing on , and . Then has a maximal fixed point and a least fixed point in such that .*

*Proof. *By Theorem 3 and Remark 5, has a maximal fixed point and hence . Set
Clearly, and hence . Define a relation on by
for all , then it is easy to check that is a partial order on .

Let be a decreasing chain of , where . From (1), (18), and (19) we find that is an increasing chain of , where
Set . Clearly, . Following the proof of Theorem 3, there exists and an increasing sequence satisfying (6) such that (8) and (11) are satisfied. From we have that for all and all . Thus the increasing property of on implies that, for all and all ,
and hence by (1),
for all and all . Let , then by (8) and the continuity of we have ; that is,
for all . This together with implies . Then in analogy to the proof of Theorem 3, by (6), (8), and we can prove has an upper bound . By (18), we have . Note that is an upper bound of in , then for all and hence by (19),
for all . This means is a lower bound of in . By Zorn's lemma, has a minimal element; denote it by . By (18) we have and
for all . By the increasing property of , we have and for all , which implies and . Moreover by (19), . The minimality of in forces that and so we have . Finally by (25), is a least fixed point of in and . The proof is complete.

Theorem 7. *Let be a complete partially ordered cone metric space over a total order minihedral and continuous cone of a normed vector space . Let be a sequentially continuous vector functional and let be a single-valued mapping. Assume that there exists such that is bounded above on , is increasing on , and . Then has a minimal fixed point and a largest fixed point in in such that .*

*Proof. *By Theorem 4 and Remark 5, has a minimal fixed point in . Set
Define a relation on as follows:
for all , then is a partial order on . In an analogy to the proof of Theorem 4, we can prove has a minimal element and is a largest fixed point of in . The proof is complete.

Theorem 8. *Let be a complete partially ordered cone metric space over a total order minihedral and continuous cone of a normed vector space . Let be a sequentially continuous mapping and let be a single-valued mapping. Assume that there exists with such that is increasing on and . Then has a largest fixed point and a least fixed point in such that .*

*Proof. *For all , by (1) we have ; that is, is bounded on . In an analogy to the proof of Theorem 3, we can prove has a maximal fixed point and a minimal fixed point in by investigating the existence of maximal element and minimal element, respectively, in and . Let
where is nonempty. Define on and on , respectively, by
then it is easy to check that and are partial orders on and , respectively. In an analogy to the proof of Theorem 4, we can prove has a minimal element and has a minimal element . By the definitions of and , we have ,
Moreover by (30) and the increasing property of on , for all , we have
and so by (31),
From (32) and (33) we have that , , and
which implies and by the minimality of and . This means that and . Hence is the least fixed point and is the largest fixed point of in by (31). The proof is complete.

*Remark 9. *Theorems 3–8 are extensions of [4, Theorems 3 and 4] and [2, Theorems 3, 4, and 5] to the case of cone metric spaces. It is worth mentioning that in Theorems 4, 7, and 8, not only the existence of maximal and minimal fixed points but also the existence of largest and least fixed points is obtained. Therefore Theorems 4, 7, and 8 are still new even in the case of metric space and hence they indeed improve [2, Theorems 3, 4, and 6].

Now we give an example to demonstrate Theorem 3.

*Example 10. *Let , with the norm for all and . Clearly, is a strongly minihedral and continuous cone of . Define a mapping by
then is a complete cone metric space over and hence is a complete cone metric subspace of . Define a vector functional by
for all . For arbitrary , let be a sequence such that , then and hence , that is, . This means that is sequentially continuous; in particular, is sequentially continuous. From (35) and (36) it is easy to check that
where is the partial order defined by (1). Let be a set-valued mapping such that
Fix , then by (37), and so . For , if and , then we have only two cases: and by (37). Fix and , for all , there exists such that . Fix and , for all , there exists such that . This means that is increasing on . Therefore all the conditions of Theorem 3 are satisfied and hence has a fixed point .

Fix ; for all , we have by (37); that is, (2) is not satisfied. Therefore the existence of fixed points could not be obtained by generalized Caristi's fixed point theorems in cone metric spaces of [1, 3].

#### Acknowledgments

The work was supported by Natural Science Foundation of China (11161022), Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), Educational Department of Jiangxi Province (GJJ12280), and Program for Excellent Youth Talents of JXUFE (201201). The authors are grateful to the editor and referees for their critical suggestions led to the improvement of the presentation of the work.

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#### Copyright

Copyright © 2013 Zhilong Li and Shujun Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.